The Molecular Hamiltonian

For a molecular system, 4 is a function of the positions f the electrons and thenu 1 within the molecule, which we will designate as r and R, respectively These h are a shorthand for the set of component vectors describing the position of each particle. We ll use subscripted versions of theig to denote the vector correspondin to a particular electron or nucleus r, andR/. Note that electrons are treated individually, while each nucleus is treated as an aggregate the component nucleon.s are not treated individually. 

The Hamiltonian is made up of kinetic and potential energy terms  

The kinetic energy is a summation of over all the particles in the molecule  

The potential energy component is the Coulomb repulsion between each pair of charged entities (treating each atomic nucleus as a single charged mass)  

The first term corresponds to electron-nuclear attraction, the second to electron-electron repulsion, and the third to nuclear-nuclear repulsion. 

In choosing this hamiltonian, we are in effect electing to seek an energy of an idealized nonexistent system— a nonrelativistic system with clamped nuclei and no magnetic moments. If we wish to make a very accurate comparison of our computed results with experimentally measured energies, it is necessary to modify either the experimental or the theoretical numbers to compensate for the omissions in H. 

The wavefunction for an SCF calculation is one or more antisymmetrized products of one-electron spin-orbitals. We have already seen (Chapter 5) that a convenient way to produce an antisymmetrized product is to use a Slater determinant. Therefore, we take the trial function to be made up of Slater determinants containing spin-orbitals (j . If we are dealing with an atom, then the (f s, are atomic spin-orbitals. For a molecule, they are molecular spin-orbitals. 

For the present, we restrict our discussion to closed-shell single-determinantal wave-functions. 

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We have alluded to the comrection between the molecular PES and the spectroscopic Hamiltonian. These are two very different representations of the molecular Hamiltonian, yet both are supposed to describe the same molecular dynamics. Furthemrore, the PES often is obtained via ab initio quairtum mechanical calculations while the spectroscopic Hamiltonian is most often obtained by an empirical fit to an experimental spectrum. Is there a direct link between these two seemingly very different ways of apprehending the molecular Hamiltonian and dynamics And if so, how consistent are these two distinct ways of viewing the molecule ... 

At this point the reader may feel that we have done little in the way of explaining molecular synnnetry. All we have done is to state basic results, nonnally treated in introductory courses on quantum mechanics, connected with the fact that it is possible to find a complete set of simultaneous eigenfiinctions for two or more commuting operators. However, as we shall see in section Al.4.3.2. the fact that the molecular Hamiltonian //coimmites with and F is intimately coimected to the fact that //commutes with (or, equivalently, is invariant to) any rotation of the molecule about a space-fixed axis passing tlirough the centre of mass of the molecule. As stated above, an operation that leaves the Hamiltonian invariant is a symmetry operation of the Hamiltonian. The infinite set of all possible rotations of the... 

The tliree protons in PH are identical aud indistinguishable. Therefore the molecular Hamiltonian will conmuite with any operation that pemuites them, where such a pemiutation interchanges the space and spin coordinates of the protons. Although this is a rather obvious syimnetry, and a proof is hardly necessary, it can be proved by fomial algebra as done in chapter 6 of [1]. 

By definition, a synnnetry operation R connnutes with the molecular Hamiltonian //and so we can write the operator equation ... 

This definition causes the wavefiinction to move with the molecule as shown for the X direction in figure Al.4,3. The set of all translation synnnetry operations / constitiites a group which we call the translational group G. Because of the imifomhty of space, G is a synnnetry group of the molecular Hamiltonian //in that all its elements commute with // ... 

The translational linear momentum is conserved for an isolated molecule in field free space and, as we see below, this is closely related to the fact that the molecular Hamiltonian connmites with all... 

Initially, we neglect tenns depending on the electron spin and the nuclear spin / in the molecular Hamiltonian //. In this approximation, we can take the total angular momentum to be N(see (equation Al.4.1)) which results from the rotational motion of the nuclei and the orbital motion of the electrons. The components of. m the (X, Y, Z) axis system are given by ... 

Since space is isotropic, K (spatial) is a symmetry group of the molecular Hamiltonian v7in that all its elements conmuite with // ... 

If we allow for the tenns in the molecular Hamiltonian depending on the electron spin - (see chapter 7 of [1]), the resulting Hamiltonian no longer connnutes with the components of fVas given in (equation Al.4.125), but with the components of... 

The Hamiltonian considered above, which connmites with E, involves the electromagnetic forces between the nuclei and electrons. However, there is another force between particles, the weak interaction force, that is not invariant to inversion. The weak charged current mteraction force is responsible for the beta decay of nuclei, and the related weak neutral current interaction force has an effect in atomic and molecular systems. If we include this force between the nuclei and electrons in the molecular Hamiltonian (as we should because of electroweak unification) then the Hamiltonian will not conuuiite with , and states of opposite parity will be mixed. However, the effect of the weak neutral current interaction force is mcredibly small (and it is a very short range force), although its effect has been detected in extremely precise experiments on atoms (see, for... 

Robb, Bemaidi, and Olivucci (RBO) [37] developed a method based on the idea that a conical intersection can be found if one moves in a plane defined by two vectors xi and X2, defined in the adiabatic basis of the molecular Hamiltonian H. The direction of Xi corresponds to the gradient difference... 

The mixing has nothing to do with the possibility of any molecules populating the term, which is typically 12,000 cm" above the ground state term. The population of such a term is of the order -12000/200 room temperature kT 200 cm" at 300 K), which is absolutely negligible. The mixing arises because a description of the molecular Hamiltonian in terms of Eq. (5.14) is incomplete and should be replaced with Eq. (5.15). 

Secondly, information is obtained on the nature of the nuclei in the molecule from the cusp condition [11]. Thirdly, the Hohenberg-Kohn theorem points out that, besides determining the number of electrons, the density also determines the external potential that is present in the molecular Hamiltonian [15]. Once the number of electrons is known from Equation 16.1 and the external potential is determined by the electron density, the Hamiltonian is completely determined. Once the electronic Hamiltonian is determined, one can solve Schrodinger s equation for the wave function, subsequently determining all observable properties of the system. In fact, one can replace the whole set of molecular descriptors by the electron density, because, according to quantum mechanics, all information offered by these descriptors is also available from the electron density. 

The transition from (1) and (2) to (5) is reversible each implies the other if the variations 5l> admitted are completely arbitrary. More important from the point of view of approximation methods, Eq. (1) and (2) remain valid when the variations 6 in a trial function are constrained in some systematic way whereas the solution of (5) subject to model or numerical approximations is technically much more difficult to handle. By model approximation we shall mean an approximation to the form of as opposed to numerical approximations which are made at a lower level once a model approximation has been made. That is, we assume that H, the molecular Hamiltonian is fixed (non-relativistic, Born-Oppenheimer approximation which itself is a model in a wider sense) and we make models of the large scale electronic structure by choice of the form of and then compute the detailed charge distributions, energetics etc. within that model. 

Before investigating the qualitative concepts of the VSEPR model it is worth noting that the details of the interactions between the electron pairs have been ascribed to a size-Pauli exclusion principle result . But objects do not repel each other simply because of their sizes (i.e. interpenetrations) only if the constituents of the objects interact is any interaction possible10). If we are to use the idea of orbital size at all we must avoid the danger of contrasting a phenomenon (electron repulsion) with one of its manifestations (steric effects). The only quantitative tests which we can apply to the VSEPR model are ones based on the terms in the molecular Hamiltonian specifically, electron repulsion. 

In semiempirical calculations variational principle itself is manipulated, or to give another interpretation, the molecular Hamiltonian is modified in order to save computation time. Two semiempirical methods differing in the concept and the degree of sophistication have been applied frequently to ion-molecule complexes and therefore will be mentioned briefly. 

Performing a partial Wigner transformation with respect to the nuclear variables, the molecular Hamiltonian can be written as... 

X + P )/4, which by construction varies between 0 (system is in /i)) and 1 (system is in /2)). Describing, as usual, the nuclear motion through the position X, the vibronic PO can then be drawn in the (A dia,v) plane. Here, the subscript dia emphasizes that we refer to the population of the diabatic states which are used to define the molecular Hamiltonian H. For inteipretational purposes, on the other hand, it is often advantageous to change to the adiabatic electronic representation. Introducing the adiabatic population A ad. where Nad = 0 corresponds to the lower and A ad = 1 to the upper adiabatic electronic state, the vibronic PO can be viewed in the (Nad,x) plane. Alternatively, one may represent the vibronic PO as a curve N d i + (1 — A ad)IFi between the... 

Another subtlety is that the assumption nuclei behave as Dirac particles, amounts to assuming that all nuclei have spin 1/2. However, it is not uncommon to have nuclei with spin as high as 9/2 worse nuclei with integer spins are bosons and do not obey Fermi-Dirac statistics. The only justification to use equation (75) for such a case is that the resulting theory agrees with experiment. Under the assumption, we are in a position to extend our many-fermion Hamiltonian to molecules assuming that the nuclei are Dirac particles with anomalous spin. The molecular Hamiltonian may then be written as... 

The symbol V(q,Q) stands for a kinematic operator containing spin-orbit terms, electron-phonon couplings and, eventually, a coupling to external fields. The molecular Hamiltonian is given by ... 

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